Convex Regularization Method for Solving Cauchy Problem of the Helmholtz Equation

Abstract

In this paper, we introduce the Convex Regularization Method (CRM) for regularizing the (instability) solution of the Helmholtz equation with Cauchy data. The CRM makes it possible for the solution of Helmholtz equation to depend continuously on the small perturbations in the Cauchy data. In addition, the numerical computation of the reg- ularized Helmholtz equation with Cauchy data is stable, accurate and gives high rate of convergence of solution in Hilbert space. Undoubtedly, the error estimated analysis associated with CRM is minimal.

Mathematics Subject Classi cation: 44B28; 44B30

Keywords: Convex Regularization Method, ill-posed Helmholtz equation with Cauchy data, stable solution